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Theorem tgbtwntriv2 27738
Description: Betweenness always holds for the second endpoint. Theorem 3.1 of [Schwabhauser] p. 30. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
tkgeom.p 𝑃 = (Baseβ€˜πΊ)
tkgeom.d βˆ’ = (distβ€˜πΊ)
tkgeom.i 𝐼 = (Itvβ€˜πΊ)
tkgeom.g (πœ‘ β†’ 𝐺 ∈ TarskiG)
tgbtwntriv2.1 (πœ‘ β†’ 𝐴 ∈ 𝑃)
tgbtwntriv2.2 (πœ‘ β†’ 𝐡 ∈ 𝑃)
Assertion
Ref Expression
tgbtwntriv2 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐡))

Proof of Theorem tgbtwntriv2
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 simprl 770 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐡 ∈ (𝐴𝐼π‘₯))
2 tkgeom.p . . . . . 6 𝑃 = (Baseβ€˜πΊ)
3 tkgeom.d . . . . . 6 βˆ’ = (distβ€˜πΊ)
4 tkgeom.i . . . . . 6 𝐼 = (Itvβ€˜πΊ)
5 tkgeom.g . . . . . . 7 (πœ‘ β†’ 𝐺 ∈ TarskiG)
65ad2antrr 725 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ 𝐺 ∈ TarskiG)
7 tgbtwntriv2.2 . . . . . . 7 (πœ‘ β†’ 𝐡 ∈ 𝑃)
87ad2antrr 725 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ 𝐡 ∈ 𝑃)
9 simplr 768 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ π‘₯ ∈ 𝑃)
10 simpr 486 . . . . . 6 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))
112, 3, 4, 6, 8, 9, 8, 10axtgcgrid 27714 . . . . 5 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)) β†’ 𝐡 = π‘₯)
1211adantrl 715 . . . 4 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐡 = π‘₯)
1312oveq2d 7425 . . 3 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ (𝐴𝐼𝐡) = (𝐴𝐼π‘₯))
141, 13eleqtrrd 2837 . 2 (((πœ‘ ∧ π‘₯ ∈ 𝑃) ∧ (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡))) β†’ 𝐡 ∈ (𝐴𝐼𝐡))
15 tgbtwntriv2.1 . . 3 (πœ‘ β†’ 𝐴 ∈ 𝑃)
162, 3, 4, 5, 15, 7, 7, 7axtgsegcon 27715 . 2 (πœ‘ β†’ βˆƒπ‘₯ ∈ 𝑃 (𝐡 ∈ (𝐴𝐼π‘₯) ∧ (𝐡 βˆ’ π‘₯) = (𝐡 βˆ’ 𝐡)))
1714, 16r19.29a 3163 1 (πœ‘ β†’ 𝐡 ∈ (𝐴𝐼𝐡))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  β€˜cfv 6544  (class class class)co 7409  Basecbs 17144  distcds 17206  TarskiGcstrkg 27678  Itvcitv 27684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-nul 5307
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-ov 7412  df-trkgc 27699  df-trkgcb 27701  df-trkg 27704
This theorem is referenced by:  tgbtwncom  27739  tgbtwntriv1  27742  tgcolg  27805  legid  27838  hlid  27860  lnhl  27866  tglinerflx2  27885  mirreu3  27905  mirconn  27929  symquadlem  27940  outpasch  28006  hlpasch  28007
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